Multiple solutions for a superlinear and periodic elliptic system on \({\mathbb{R}^N}\). (English) Zbl 1279.35038
In this paper, the authors consider a periodic Hamiltonian elliptic system where \(0\) lies in a gap of the spectrum to the associated Schrödinger operator. To treat the problem variationally, they build a suitable functional framework taking into account the polar decomposition and the fractional powers of the Schrödinger operator. By adopting the monotonicity trick due to Jeanjean, they show the existence of a ground state solution and the multiplicity of solutions to the initial system.
Reviewer: Mohammed Bouchekif (Tlemcen)
References:
| [1] | Alves C.O., Carrião P.C., Miyagaki O.H.: On the existence of positive solutions of a perturbed Hamiltonian system in $${\(\backslash\)mathbb{R}\^N}$$ . J. Math. Anal. Appl. 276, 673–690 (2002) · Zbl 1056.35060 · doi:10.1016/S0022-247X(02)00413-4 |
| [2] | Ávila A.I., Yang J.: On the existence and shape of least energy solutions for some elliptic systems. J. Diff. Eqns. 191, 348–376 (2003) · Zbl 1109.35325 · doi:10.1016/S0022-0396(03)00017-2 |
| [3] | Bartsch T., De Figueiredo D.G.: Infinitely Many Solutions of Nonlinear Elliptic Systems, Progress in Nonlinear Differential Equations and Their Applications, vol. 35, pp. 51–67. Birkhäuser, Basel/Switzerland (1999) · Zbl 0922.35049 |
| [4] | Bartsch T., Ding Y.: Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math. Nach. 279, 1–22 (2006) · Zbl 1117.58007 · doi:10.1002/mana.200410420 |
| [5] | Benci V., Rabinowitz P.H.: Critical point theorems for indefinite functionals. Inven. Math. 52, 241–273 (1979) · Zbl 0465.49006 · doi:10.1007/BF01389883 |
| [6] | Coti Zelati V., Rabinowitz P.H.: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Am. Math. Soc. 4, 693–727 (1991) · Zbl 0744.34045 · doi:10.2307/2939286 |
| [7] | De Figueiredo D.G., Ding Y.: Strongly indefinite functionals and multiple solutions of elliptic systems. Tran. Am. Math. Soc. 355, 2973–2989 (2003) · Zbl 1125.35338 · doi:10.1090/S0002-9947-03-03257-4 |
| [8] | De Figueiredo D.G., Felmer P.L.: On superquadratic elliptic systems. Tran. Am. Math. Soc. 343, 97–116 (1994) · Zbl 0799.35063 |
| [9] | De Figueiredo D.G., DO Ó B., Ruf J.M.: An Orlicz-space approach to superlinear elliptic systems. J. Func. Anal. 224, 471–496 (2005) · Zbl 1210.35081 · doi:10.1016/j.jfa.2004.09.008 |
| [10] | De Figueiredo D.G., Yang J.: Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Anal. 33, 211–234 (1998) · Zbl 0938.35054 · doi:10.1016/S0362-546X(97)00548-8 |
| [11] | Gohberg I., Goldberg S., Kaashoek M.A.: Classed of Linear Operators, vol. I. Birkhäuser-Verlag, Basel, Boston, Berlin (1990) |
| [12] | Hulshof J., Van De Vorst R.C.A.M.: Differential systems with strongly variational structure. J. Func. Anal. 114, 32–58 (1993) · Zbl 0793.35038 · doi:10.1006/jfan.1993.1062 |
| [13] | Jeanjean L.: On the existence of bounded Palais-Smale sequences and application to a Landesmann-Lazer type problem set on $${\(\backslash\)mathbb{R}\^N}$$ . Proc. R. Soc Edinb. 129 A, 787–809 (1999) · Zbl 0935.35044 · doi:10.1017/S0308210500013147 |
| [14] | Kato T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin, Heidelberg, New York (1976) · Zbl 0342.47009 |
| [15] | Kryszewski W., Szulkin A.: An infinite dimensional Morse theory with applications. Tran. Am. Math. Soc. 349, 3181–3234 (1997) · Zbl 0892.58015 · doi:10.1090/S0002-9947-97-01963-6 |
| [16] | Li G., Yang J.: Asymptotically linear elliptic systems. Comm. Partial Diff. Eqns. 29, 925–954 (2004) · Zbl 1140.35406 · doi:10.1081/PDE-120037337 |
| [17] | Li Y., Wang Z., Zeng J.: Ground states of nonlinear Schrödinger equations with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 23, 829–837 (2006) · Zbl 1111.35079 · doi:10.1016/j.anihpc.2006.01.003 |
| [18] | Lions P.L.: The concentration compactness principle in the calculus of variations. The locally compact case. Part II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 223–283 (1984) · Zbl 0704.49004 |
| [19] | Pistoia A., Ramos M.: Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions. J. Diff. Eqns. 201, 160–176 (2004) · Zbl 1246.35089 · doi:10.1016/j.jde.2004.02.003 |
| [20] | Reed M., Simon B.: Methods of Modern Mathematical Physics, IV Analysis of Operators. Academic Press, New York (1978) · Zbl 0401.47001 |
| [21] | Séré E.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209, 133–160 (1992) · doi:10.1007/BF02570817 |
| [22] | Sirakov B.: On the existence of solutions of Hamiltonian elliptic systems in R N . Adv. Diff. Eqns. 5, 1445–1464 (2000) · Zbl 1213.35223 |
| [23] | Sirakov B., Soares S.H.M.: Soliton solutions to systems of coupled Schrödinger equations of Hamiltonian type. Trans. Am. Math. Soc. 362, 5729–5744 (2010) · Zbl 1202.35081 · doi:10.1090/S0002-9947-2010-04982-7 |
| [24] | Szulkin A., Weth T.: Ground state solutions for some indefinite problems. J. Funct. Anal. 257, 3802–3822 (2009) · Zbl 1178.35352 · doi:10.1016/j.jfa.2009.09.013 |
| [25] | Szulkin A., Zou W.: Homoclinic orbits for asymptotically linear Hamiltonian systems. J. Funct. Anal. 187, 25–41 (2001) · Zbl 0984.37072 · doi:10.1006/jfan.2001.3798 |
| [26] | Triebel H.: Interpolation Theory, Function Spaces, Differential Operators. Amsterdam, North-Holland (1978) · Zbl 0387.46032 |
| [27] | Van der Vorst R.C.A.M.: Variational identities and applications to differential systems. Arch. Ration. Mech. Anal. 116, 375–398 (1991) · Zbl 0796.35059 · doi:10.1007/BF00375674 |
| [28] | Zhao F., Zhao L., Ding Y.: Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems. ESAIM Control Optimisation Calc. Var. 16, 77–91 (2010) · Zbl 1189.35091 · doi:10.1051/cocv:2008064 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
